Finite volumes schemes preserving the low Mach number limit for the Euler system

Transcription

1 Finite volumes schemes preserving the low Mach number limit for the Euler system M.-H. Vignal Low Velocity Flows, Paris, Nov. 205 Giacomo Dimarco, Univ. de Ferrara, Italie Raphael Loubere, IMT, CNRS, France Funding : ANR MOONRISE

2 Outline General contet : multi-scale models and principle of s 2 The isentropic Euler system in the low Mach number regime Classical and s in the low Mach number limit 4 Numerical results 5 Works in progress

3 electron density (log scale) : t= General contet Multiscale model : M ε depends on a parameter ε In the (space-time) domain ε can be small compared to the reference scale be of same order as the reference scale take intermediate values ε When ε is small : M 0 = lim ε 0 M ε asympt. model Difficulties : Classical eplicit schemes for M ε : stable and consistent iff the mesh resolved all the scales of ε huge cost Schemes for M 0 with meshes independent of ε But : M 0 not valid everywhere, needs ε location of the interface, moving interface

4 Principle of s A possible solution : s Use the multi-scale model M ε where you want. Discretize it with a scheme preserving the limit ε 0 The mesh is independent of ε : Asymptotic stability. You recover an approimate solution of M 0 when ε 0 : Asymptotic consistency Asymptotically stable and consistent scheme Asymptotic preserving scheme (AP) ([S.Jin] kinetic hydro) You can use the only to reconnect M ε and M 0 M ε class. scheme ε=o() intermediate zone ε M 0 class. scheme M ε

5 Outline General contet : multi-scale models and principle of s 2 The isentropic Euler system in the low Mach number regime Classical and s in the low Mach number limit 4 Numerical results 5 Works in progress

6 The multi-scale model and its asymptotic limit Isentropic Euler system in scaled variables Ω IR d, t 0 { t ρ+ (ρu)= 0, () ε (M ε ) t (ρu)+ (ρu u)+ ε p(ρ)=0, (2) ε Parameter : ε=m 2 = m u 2 /(γp(ρ)/ρ), M =Mach number Boundary and initial conditions : { ρ(,0)=ρ0 + ε ρ 0 (), u n=0, on Ω, and The formal limit ε 0 u(,0)=u 0 ()+εũ 0 (), with u 0 = 0. (2) ε p(ρ)=0, ρ(,t)=ρ(t). () ε Ω ρ (t)+ρ(t) u n=0, ρ(t)=ρ(0)=ρ 0, u= 0 Ω

7 The multi-scale model and its asymptotic limit The asymptotic model : Rigorous limit [Klainerman, Majda, 8] ρ=cste=ρ 0, (M 0 ) ρ 0 u = 0, () 0 ρ 0 t u+ ρ 0 (u u)+ π = 0, (2) 0 where ( ) π = lim p(ρ) p(ρ 0 ). ε 0 ε Eplicit eq. for π t () 0 (2) 0 The pressure wave eq. from M ε : π = ρ 0 2 :(u u). t () ε (2) ε tt ρ ε p(ρ)= 2 :(ρu u) () ε From a numerical point of view Eplicite treatment of () ε conditional stability t ε Implicite treatment of () ε uniform stability with respect to ε

8 Bibliography All speed schemes Preconditioning methods : [Chorin 65], [Choi, Merkle 85], [Turkel, 87], [Van Leer, Lee, Roe, 9], [Li,Gu 08,0], Splitting and pressure correction : [Harlow, Amsden, 68,7], [Karki, Patankar, 89], [Bijl, Wesseling, 98], [Sewall, Tafti, 08], [Klein, Botta, Schneider, Munz, Roller 08], [Guillard, Murrone, Viozat 99, 04, 06] [Herbin, Kheriji, Latché 2,], Asymptotic preserving schemes [Degond, Deluzet, Sangam, V, 09], [Degond, Tang ], [Cordier, Degond, Kumbaro 2], [Grenier, Vila, Villedieu ] [Dellacherie, Omnes, Raviart,], [Noelle, Bispen, Arun, Lukacova, Munz,4], [Chalons, Girardin, Kokh,5] [Dimarco, Loubère, V, in prep.]

9 Outline General contet : multi-scale models and principle of s 2 The isentropic Euler system in the low Mach number regime Classical and s in the low Mach number limit 4 Numerical results 5 Works in progress

10 Classical scheme in the low Mach number limit If ρ n and u n are known at time t n ρ n+ ρ n + (ρu) n = 0, t () n (ρu) n+ (ρu) n + (ρu u) n + t ε p(ρn )=0. (2) Reformulation () n+ () n t (2) ρ n+2 2ρ n+ + ρ n t 2 ε p(ρn )= 2 :(ρu u) n Eplicit treatment conditional stability t ε ε 0 gives p(ρ n )=0 consistency lost at the limit

11 in the low Mach number limit Goal : Build an for the multi-scale model M ε Uniform stability in ε t = O() Consistency for all ε 0 : ε 0 scheme for the limit model M 0 Basic ideas : Implicit treatment of the right terms Reformulation to make appear the limit model, Euler-Lorentz, low Mach number and drift limits [P. Degond, F. Deluzet, A. Sangam, MHV, JCP 2009] for Euler, low Mach number limit [P. Degond, M. Tang, Commun. Comput. Phys, 20] Parameter α fied by the user, depends on the considered problem If ε then α small If ε=o() then α sufficiently large Etension to the full Euler and Navier-Stokes systems : [ F. Cordier, P. Degond, A. Kumbaro, JCP 202]

12 in the low Mach number limit Our : If ρ n and u n are known at time t n ρ ρ n + (ρu) n+ = 0, t () n (ρu) n+ (ρu) n + (ρu u) n + t ε p(ρn+ )=0. (2) Reformulation () n () n t (2) ρ n+ 2ρ n + ρ n t 2 ε p(ρn+ )= 2 :(ρu u) n Implicit treatment uniform stability in ε ε 0 gives p(ρ n+ )=0 consistency at the limit

13 Uncoupled formulation of the Take the divergence of the momentum eq. (2) (2) (ρu) n+ = (ρu) n t 2 :(ρu u) n t ε p(ρn+ ) Insert it in the mass eq. () ρ n+ ρ n t + (ρu) n t ε p(ρn+ ) t 2 :(ρu u) n = 0. Uncoupled formulation of the ρ n+ ρ n + (ρu) n t t ε p(ρn+ ) t 2 :(ρu u) n = 0, (ρu) n+ (ρu) n + (ρu u) n + t ε p(ρn+ )=0. Space discretization : Need a correct choice of the numerical viscosity

14 -D space discretization for clarity Splitting : W n+ W n (0,ρu 2 ) n (ρu, p(ρ)/ε) {}}{{}} n+ { + F e (W n )+ F i (W n+ ) = 0 t }Mj j /2 j /2 j+/2 j+/2 W n+ j Wj n + f e(wj n,wj+ n ) f e(w n t j,wn j ) + f i(w n+ j,w n+ j+ ) f i(w n+ j,wn+ j ) = 0

15 -D space discretization for clarity Rusanov flues f e (Wj n,wj+ n )= F e(wj n )+F e (Wj+ n ) De n 2 (W j n Wj+ n ). f i (W n+ j,w n+ j+ )= F i(w n+ j )+F i (W n+ j+ ) D n+ i (W n+ j W n+ j+ 2 ). Jacobian matrices ( 0 0 DF e (W)= u 2 2u ) and DF i (W)= ( 0 c 2 /ε 0 ) Eigen values 0, 2u c/ ε, c/ ε with c 2 = p (ρ)

16 Stability of the linearized system around(ρ 0,u 0 ) Under the CFL Eplicit viscosity t 2 u 0 2D n e = 2 u 0 Implicit viscosity If D n+ i = 0 L 2 stability result If 2D n+ i = c 0 / ε L stability result

17 Uncoupled scheme in the non linear case Reformulation : W n+ W n (0,(ρu 2 ) n ) ((ρu) n, p(ρ n+ )/ε) {}}{{}}{ + F e (W n )+ F i (W n+/2 ) t ( t p(ρ n+ ) ε +(ρu 2 ) n 0 ) = 0 Step Determine ρ n+ j knowing W n =(ρ n,(ρu) n ) ρ n+ j ρ n j t + (ρu)n j +(ρu) n j+ 2 Dn e+ D n i dis (ρ n ) {}}{ (ρ n j+ 2ρn j + ρ n j ) t ε disp(ρ n+ ) t dis (ρu 2 ) n = 0.

18 Uncoupled scheme in the non linear case Step 2 Determine(ρu) n+ j knowing V n+/2 =(W n,ρ n+ ) (ρu) n+ j (ρu) n j t Flues given by + n+/2 g(vj,v n+/2 j+ ) g(v n+/2 j,v n j ) = 0 g(v n+/2 j,v n+/2 j+ )= (ρu2 ) n j + p(ρ n+ j )/ε+(ρu 2 ) n j+ + p(ρn+ j+ )/ε 2 C.F.L Condition : (D n e + Dn i )((ρu) n j (ρu) n j+ ). stable if t ma( 2u n j ) = O()

19 Outline General contet : multi-scale models and principle of s 2 The isentropic Euler system in the low Mach number regime Classical and s in the low Mach number limit 4 Numerical results 5 Works in progress

20 -D test case Domain[0, ], = /500 or /00 Final time : T = 0.05, Pressure law p(ρ)=ρ γ, γ=2.00 +ε ε Initial density +ε/2 ε/ Initial momentum

21 Non linear case γ=2 and ε=o(), = /500 ε=0.99, D i = 0 Class : 27 loops CPU time 0.07 AP : 50 loops CPU time.46 Time steps Time Density Momentum

22 Non linear case γ=2 and ε=0.99, = / AP D i = AP D i 0 Density.2 Density Momentum AP D i = 0 Momentum AP D i D i = 0 centered implicit scheme D i 0 upwind implicit scheme

23 Non linear case γ=2 and ε ε=0 4 0 Class : 406 loops CPU time 0.82 AP : 57 loops CPU time 0.4 Time steps Time Density Momentum

24 Non linear caseγ=2 and ε ε=0 4 Underlying of the viscosity Time steps Time Density Momentum

25 Same test case γ=2, ε=ε(t), = / Class : 6906 loops CPU time 6.62 AP : 4 loops > 60 CPU time 2. > 55 ε(t) Time plot Time steps Density at t = Time If T final = : Class : 400 loops, CPU time 22.9 AP : 4 loops, CPU time 2.47 > 0

26 ε=ε(t), = / plot ε(t) plot plot Density at t = Time Density at t = Density at t = T final = : Class : 4087 loops, CPU time 95.8 AP : 448 loops, CPU time > 9.

27 Non linear case γ=2, zero velocity ε=0.99 Same initial density But zero initial momentum Time steps Time Momentum Momentum

28 Outline General contet : multi-scale models and principle of s 2 The isentropic Euler system in the low Mach number regime Classical and s in the low Mach number limit 4 Numerical results 5 Works in progress

29 Full Euler system Domain[0,], = /500 Final time : T = 0.05, γ=.4, ρ(t = 0)=, p(t = 0)= ε/2 ε/ Initial momentum Initial total energy

30 Full Euler system, ε=0.99 Class : 75 loops, CPU 0.02 AP : 67 loops, CPU Time steps Density Time Momentum Total energy

31 Full Euler system, ε=0 4 Class : 629 loops, CPU 0.86 AP : 45 loops, CPU Time steps Time Density Momentum Total energy

32 Full Euler system, ε=0 4 Underlying of the viscosity, 000 cells 0 5 Time steps Density Time Momentum Total energy

33 2D results in progress, γ= vnu_0.gnu vnu_00.gnu Initial density and zero velocity Classical density ε=0.99 vnu_00.gnu AP density ε= For ε=0.0, Class. 80 loops, CPU AP 89 loops, CPU 29. >.5

34 Works in progress Analysis of the, convergence in the low-mach number regime, staggered grids [Herbin, Latché, Saleh, V, in preparation] High order space and time algorithms preserving the AP properties, multi-dimensional test cases in physical variables (ε=ε()) Initial and boundary conditions not well prepared to the low Mach number regime, [Métivier, Schochet, 200], [Alazard, 2005] Use the only in the intermediate zones Domain decomposition